Problem: $ 2.\overline{37} \div 2.\overline{1} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 237.3737...\\ x &= 2.3737...\end{align*} $ $\begin{align*} 99x &= 235 \\ x &= \dfrac{235}{99}\end{align*} $ $\begin{align*} 10y &= 21.1111...\\ y &= 2.1111...\end{align*} $ $\begin{align*} 9y &= 19 \\ y &= \dfrac{19}{9}\end{align*} $ So, the problem becomes: $ \dfrac{235}{99} \div \dfrac{19}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{235}{99} \times \dfrac{9}{19} = {?} $ $ \phantom{\dfrac{235}{99} \times \dfrac{19}{9}} = \dfrac{235 \times 9}{99 \times 19} $ $ \phantom{\dfrac{235}{99} \times \dfrac{19}{9}} = \dfrac{235 \times \cancel{9}} {\cancel{99}11 \times 19} $ $ \phantom{\dfrac{235}{99} \times \dfrac{19}{9}} = \dfrac{235}{209} $